Searcharxiv⌕ Search

arXiv subjects

Sergei Kalmykov

Publications and source records attributed to Sergei Kalmykov.

17 recordsLinked to original sources
arXiv2026

Strong asymptotics for Jacobi-Piñeiro orthogonal polynomials

We investigate the asymptotic behavior of Jacobi-Piñeiro polynomials of degree $2n$ orthogonal on $[0,1]$ with respect to weights $w_j(x) = x^{α_j}(1-x)^β$, $j=1,2$ where $α_1,α_2, β>-1$, and $α_1-α_2 \notin \mathbb{Z}$. These polynomials are characterized by a Riemann-Hilbert problem for a $3 \times 3$ matrix-valued function. We use the Deift-Zhou steepest descent method for Riemann-Hilbert problems to obtain strong uniform asymptotics in the complex plane. The local parametrix around the origin is constructed using Meijer G-functions. We match the local parametrix around the origin with the global parametrix with a double matching, a technique that was recently introduced.

math.CA↗
arXiv2026

Evaluation of terminating and non-terminating sums containing the digamma function

We derive transformation and summation formulas for terminating and nonterminating series involving the digamma function. Our principal results are obtained by a limiting process starting with duality relations for the generalized hypergeometric functions and their consequences. Selected formulas are further extended by parameter differentiation of Euler's transformation and by using contiguous relations. Most of our identities express products of hypergeometric and digamma series in terms of hypergeometric functions, and some evaluations of terminating digamma sums involve Bernoulli polynomials. In several cases the digamma contributions cancel, producing identities involving only products of hypergeometric functions.

math.CA↗
arXiv2023

Minimal degree rational open up mappings and related questions

We establish the existence and uniqueness of rational conformal maps of minimal degree $n+1$ for opening up $n$ arcs. In earlier results, the degree was exponential in $n$. We also discuss two related problems. (a) We establish existence of rational functions of minimal degree with prescribed critical values, and show that the number of (suitably normalized) rational functions is given in terms of the Hurwitz numbers. (b) We consider the problem of finding rational functions of minimal degree with prescribed critical points, where we establish existence of solutions by considering certain polynomial equations, and where the number of normalized solutions is bounded from above by a Catalan number. We illustrate our results with two examples.

math.CV↗
arXiv2021

Bernstein- and Markov-type inequalities

This survey discusses the classical Bernstein and Markov inequalities for the derivatives of polynomials, as well as some of their extensions to general sets.

math.CV↗
arXiv2021

Continuity of logarithmic capacity

We prove the continuity of logarithmic capacity under Hausdorff convergence of uniformly perfect planar sets. The continuity holds when the Hausdorff distance to the limit set tends to zero at sufficiently rapid rate, compared to the decay of the parameters involved in the uniformly perfect condition. The continuity may fail otherwise.

math.CV↗
arXiv2017

Removable sets for intrinsic metric and for holomorphic functions

We study the subsets of metric spaces that are negligible for the infimal length of connecting curves; such sets are called metrically removable. In particular, we show that every totally disconnected set with finite Hausdorff measure of codimension 1 is metrically removable, which answers a question raised by Hakobyan and Herron. The metrically removable sets are shown to be related to other classes of "thin" sets that appeared in the literature. They are also related to the removability problems for classes of holomorphic functions with restrictions on the derivative.

math.CV↗
arXiv2017

Uniform convergence of Green's functions

Given a sequence of regular planar domains converging in the sense of kernel, we prove that the corresponding Green's functions converge uniformly on the complex sphere, provided the limit domain is also regular, and the connectivity is uniformly bounded.

math.CV↗
arXiv2016

Bernstein- and Markov-type inequalities for rational functions

Asymptotically sharp Bernstein- and Markov-type inequalities are established for rational functions on $C^2$ smooth Jordan curves and arcs. The results are formulated in terms of the normal derivatives of certain Green's functions with poles at the poles of the rational functions in question. As a special case (when all the poles are at infinity) the corresponding results for polynomials are recaptured.

math.CV↗
arXiv2016

Bernstein type inequalities for rational functions on analytic curves and arcs

Borwein and Erdélyi proved a Bernstein type inequality for rational functions on the unit circle and on the real line. Here we establish asymptotically sharp extensions of their inequalities for rational functions on analytic Jordan arcs and curves. In the proofs key roles are played by Borwein-Erdélyi inequality on the unit circle, Gonchar-Grigorjan type estimate of the norm of holomorphic part of meromorphic functions and Totik's construction of fast decreasing polynomials.

math.CA↗
arXiv2016

Extremal decomposition problems for p-harmonic radius

We extend classical results by Lavrent'ev and Kufarev concerning the product of the conformal radii of planar non-overlapping domains. We also extend relatively recent results for the case of domains in the $n$-dimensional Euclidean space, $n \geq 3$, with conformal radii replaced by harmonic ones. Namely, we get analogues of these results in $n$-dimensional Euclidean space in terms of $p$-harmonic radius. The proofs are based on technique of modulii of curve families and dissymmetrization of such families.

math.CV↗
arXiv2015

Extremal decomposition problems in the Euclidean space

Composition principles for reduced moduli are extended to the case of domains in the $n$-dimensional Euclidean space, $n>2$. As a consequence analogues of extremal decomposition theorems of Kufarev, Dubinin and Kirillova in the planer case are obtained.

math.CV↗
arXiv2015

Asymptotically sharp Markov and Schur inequalities on general sets

Markov's inequality for algebraic polynomials on $\left[-1,1\right]$ goes back to more than a century and it is widely used in approximation theory. Its asymptotically sharp form for unions of finitely many intervals has been found only in 2001 by the third author. In this paper we extend this asymptotic form to arbitrary compact subsets of the real line satisfying an interval condition. With the same method a sharp local version of Schur's inequality is given for such sets.

math.CV↗
arXiv2015

Polynomial and rational inequalities on Jordan arcs and domains

In this paper we prove an asymptotically sharp Bernstein-type inequality for polynomials on analytic Jordan arcs. Also a general statement on mapping of a domain bounded by finitely many Jordan curves onto a complement to a system of the same number of arcs with rational function is presented here. This fact, as well as, Borwein-Erdélyi inequality for derivative of rational functions on the unit circle, Gonchar-Grigorjan estimate of the norm of holomorphic part of meromorphic functions and Totik's construction of fast decreasing polynomials play key roles in the proof of the main result.

math.CV↗